All known cockroaches are omnivorous (this means they eat nearly everything) though in captivity most species do well on a mixture of dried feed/grains etc. and fresh vegetables or fruit, though a lot of people find they have success feeding them dog food as well. Rolled oats and fruit like bananas and apples seem to do well, though they especially like over-ripe peaches and plums. Though they don't need fresh food every day it is important that they always have enough to eat, otherwise they will start eating the cage as well as each other.
Housing cockroaches in captivity poses two main problems. First, most species come from tropical countries and therefore need to be kept, at least 25°C to be happy. Second, there is the problem of the cockroaches escaping and infesting the house. This problem is easily overcome by careful selection of species and cage structure, the cage needs to be escape-proof. As most species of cockroach are good climbers, a tight-fitting lid is required with a good quality fine mesh to allow for air flow. For smaller and faster moving species access can be obtained via a cloth or mesh sleeve which can be tied tightly closed when not in use.
The floor of the cage should be covered with wood shavings for leaf litter species such as Gromphadorhina portentosa or with peat for burrowing species such as Pycnoselus surinamensis as well as a mixture of surface objects to offer hiding spaces (most cockroaches are nocturnal). Egg boxes and the rolls from toilet paper or paper towels are good for this.
Heating & Lighting Requirements
The best method is to house them in a specially heated room, but for most people this is not possible, another alternative is to keep the cage in an airing cabinet. The most common method is to use a heat pad, these come in a variety of shapes and sizes. It is best to talk to your local seller as to what you need. The pad should rest comfortably under the cage/aquarium so that two thirds of the cage is directly over the pad. This allows a gradient of heat to arise giving the cockroaches some choice over what temperature they would like to be in. The use of a thermostat can make it easier but it isn't really necessary for most of the commonly kept species.
Be very careful when handling, if a few escape they can reproduce rapidly in your home. Wash hands before and after handling.
As a general rule, breeding will take care of itself. Eggs are normally laid in an ootheca (nature's answer to the polystyrene egg box). Some species will secrete these in the corners of the cage or other accessible nooks and crannies while others will carry the ootheca around with them, either inside or partly extruded from the body until it is time for the young to hatch. Some species are parthenogenic (the females give birth to other females only, etc. and no males are ever seen) i.e. Pycnoscelus surinamensis.
There are nearly 4,000 species of cockroaches (Dictyoptera, Blattodea) in the world, of which only 25 to 30 (or less than 1%) have any pest status, the rest are innocent members of the earth's fauna, some of which make great pets.